Editorial Team - Mathematical Association 21.2.pdf · In this article Mark Pepper shares his...

Vol. 21 No. 2Autumn 2016 1

Editorial Team:

Carol Buxton Lucy Cameron Mary ClarkAlan EdmistonPeter JarrettMark Pepper

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The views expressed in this journalare not necessarily those of TheMathematical Association. The inclusionof any advertisements in this journaldoes not imply any endorsement by TheMathematical Association.

Editors’ page 2

Assessment without levels 3Louise Gowan has been teaching in a special school for many years. As part of a series of articles she shares the highs and lows of working with those for whom this publication was devised.

The Curious Case of the Dog in the Night-Time (Re-visited 5after 12 years) In this article Mark Pepper shares his mathematical reflections on a most curious book.

Maths talk 9This edition of Maths Talk features Nichola, a hospital teacher who left mainstream primary teaching 20 years ago.

The teaching and learning of mathematics to pupils with 10a visual impairment Mark Pepper has a passion for inclusion, and in the first of a series of articles he outlines the difficulties facing VI students and shares a few strategies their teachers can use to support their learning.

Mikos De Symmetria: a special approach to Trigonometry 15We were approached by a young mathematician from India, Sameer Sharma, who sent a really interesting article on Trigonometry. We felt that we could do little else but showcase the thinking he has been doing and if you would like to share any thoughts with him please e-mail the Editor who will pass them on.

Objects to think with. Papert (1980) 19In this piece Mary Clarke reflects upon the importance of the work of Papert and how his ideas might help classroom teachers today.

Review - Rachel Gibbons 22

mailto:jcpadvertising%40yahoo.co.uk?subject=Equals%20advertising%20enquiry

Vol. 21 No. 2 Autumn 20162

Editor’s Page

It gives me great pleasure to share this issue of

Equals with you.

I would like to draw your attention to the article by

Sameer Sharma, a 14-year-old student from India.

He is an incredibly keen mathematician and it is a

real privilege to publish his work. This issue’s Maths

Talk is with a Nichola, a hospital teacher, and the

similarities in her advice and that of Lynda Maples

is quite striking. Mark Pepper has supplied two

pieces for us; one on visual impairment and another

reflecting his thoughts upon a most curious read.

Rachel Gibbons has kindly reviewed Dylan Wiliam’s

key text Embedding Formative Assessment for us.

Louise Gowan continues to share her experiences

from life in a special school while Mary Clarke

highlights the importance of concrete physical

experience in the learning of mathematics.

Moving forward

The last six months have been very interesting for

Equals. A real sense of direction, and purpose, has

emerged and I am happy that Equals has a way

forward after the successful stewardship of Ray

Gibbons.

We have received a number of contacts and these

have shaped both my own thinking and the future

direction of this publication. The most significant

one came in the form of an email and phone call

from Barbara Rodgers of the Solent Maths Hub.

Barbara got in touch to ask for help with a SEND

conference. The conference is seeking to provide

support and guidance for mainstream pupils

who are struggling and attaining well below that

expected for their age. This was also coupled with

an invite to a SEND meeting to which all Maths

Hubs were invited. I am pleased to report that Lucy

Cameron, who was there to represent the Great

North Maths Hub, was able to speak on our behalf.

The outcome of the meeting was an overwhelming

sense of purpose in increasing the focus upon

SEND pupils among the Hub’s. It was felt that

Equals can play a key role in this network and that a

forum needs to exist to share ideas and network. I

feel that Equals is ideally positioned to shine a light

upon the good practice that is taking place in special

schools. Many issues were discussed such as the

importance of accreditation for older students,

while there is also a pressing need to share what

‘mastery’ might look like for SEND pupils.

I am also pleased to report that the support for

Equals is strong enough to have an editorial board

meeting. A detailed summary of the outcomes

of this meeting, which will shape the direction of

Equals, will be reported in the next edition.


Assessment without levels

20 years ago I spoke at NUT conference against

the National Curriculum and testing, outlining the

dangers of a narrowing of the curriculum and a move

toward teaching to test. This has happened but the

emphasis on levels has had further consequences.

Levels have been problematic. Children have

become defined by their levels and use levels to

describe their achievement and progress, saying ‘I

am a level 4a’ rather than talking about actual skills

they have learnt or work they have completed. Work

on walls is often levelled and some walls have levels

with names against them to show where pupils are.

All of this is a misuse of levels where a simple number

is used to describe a pupils’ achievement in Maths.

For pupils such as ours

who achieve below that

expected for their age

the whole system is very

de-moralising and de-motivating.

Levels have been used to create or encourage

competition between pupils, teachers and schools.

They have been used to create unrealistic targets

which become the main aim of the pupil, teacher

and school. We have moved towards teaching by

number, teaching toward numerical targets which

then determine our success as teachers, our pupils’

success as learners and our school’s success in

OFSTED with the threat of failure forever nearby.

As levels have become the main purpose of

education there has been a move back to teach to

test, learning by rote, exam practice for a large part

of the year. Schools have become exam factories

with success governed by test results. The effect

on the curriculum has been an emphasis on the

’core subjects’-Maths, English and Science and

a narrowing of the curriculum with creativity and

investigation being the inevitable loss within this

framework.

Everything I worried about 20 years ago has come

to be.

As a SMILE teacher of 30+ years I have always

used levels. They were

a guide to where a pupil

was and gave a starting

point for what work to set.

In SMILE every task was levelled and recorded on

a grid which showed every task completed by an

individual pupil, and recorded their success through

a short test on each task following completion of a

matrix of 10 tasks. It was a very thorough system.

Although levels were used consistently they were

never the main focus. They were useful and used

as a guide but not restrictive.

Levels are a useful way to compare pupils against

each other and to assess individual progress.

Pupils should know where they are in their learning

Louise Gowan has been teaching in a special school for many years. As part of a series of articles she shares the highs and lows of working with those for who this publication was devised.

Schools have become exam factories with success governed by test results.


and what they need to learn next. Levels can be

used as a hierarchical structure or container with

the mathematic content being held within it and

providing the food, the main purpose of the learning.

The problem has been the use of publicised results

which set pupil against pupils, teachers against

teachers and schools against schools and set

higher and higher targets. This has led to unrealistic

and very naïve interpretations of achievements,

averages and tests. Levels are a way to track

progress which is useful to both teacher and pupil

to share achievements if used appropriately.

Assessment without levels is not what is actually

happening. A variety of systems will now be put

into place which is likely to make it more difficult

for parents to understand how their children are

doing and potentially more confusing too for pupils

and teachers. The debate

continues around the

level of detail that is useful

in describing what a child

knows and how this

relates to learning intentions and success criteria.

Some systems will replace numbers with statements

and perhaps even use some qualitative rather than

quantitative data. There may be good systems

being developed but some systems are going to be

even worse than that which they replace. In some

instances pupils are now being assessed against

age appropriate ‘fundamentals’, very similar to the

key objectives of the original National Strategy.

Instead of being a 4a they will now presumably talk

about being at, below or above age appropriate.

My year 8 class are working at reception/year 1. Is

it better to talk about them working at year 1 rather

than being on a 1b? Pupils are being assessed

according to the percentage of fundamentals for a

year that they have covered. For example a pupil

having been assessed as understanding 25%

of the fundamental statements as year 2 may be

assessed as 1.025 as if a descriptor with 3 decimal

places is a more accurate assessment of progress.

While there is an attempt to reassure teachers that

their professional judgement is important and that

a variety of evidence should be used to make a

judgement my guess is that standardised tests will

be used to confirm judgements but maybe I have

become overly cynical over the years.

Some schools are, I know, using this opportunity

to be more creative in their assessment and this

has to be a good thing. Discussion and debate

in education is important and I hope that this will

be used as an opportunity for some education

professionals to take back the initiative and develop

assessments that make

sense and that can

replace the old system of

levels successfully.

I can’t remember now which secretary of state for

education started the notion of everyone being

above average but this misconception persists.

The abandonment of levels had some good

theory behind it but in effect we are still working to

making sure all achieve above average. Only the

high achievers will feel good, and even there the

pressure is to be forever better. The pressure is on

to continue to improve. A is not good enough for

the ‘top’ stream, outstanding is not good enough

for the ‘top’ schools. We have always to look

beyond ‘what went well?’ to ‘even better if’. A* and

world class are now common targets for many. It

would be nice if we could get the message across

that we will never all be above average and actually

A is not good enough for the ‘top’ stream, outstanding is not good

enough for the ‘top’ schools.


The Curious Case of the Dog in the Night-Time (Re-visited after 12 years)

In this article Mark Pepper shares his mathematical reflections on a most curious book.

In 2004, shortly after this book was first published,

Nick Peacey wrote a review of it which was published

in Equals. Nick gave the book fulsome praise and

dealt mainly with the general aspects of the effects

of Asperger Syndrome. I recently read the book for

the first time and largely agree with Nick’s sentiments

but it will now be considered from the perspective

of its mathematical content. The book is written in

the first person by Christopher, a 15 year old who

has Asperger Syndrome. As a consequence of his

condition he interprets everyday events in a highly

unorthodox manner and yet he has huge reservoirs

of factual knowledge and he is presented as having

outstanding mathematical ability.

There is plenty of evidence supporting the

proposition that Christopher is an outstanding

mathematician. This includes an elegant algebraic

proof and a sophisticated interpretation of a

complex probability question. He also performs

scarcely credible feats of mental dexterity in cubing

all of the numbers up to and beyond 17, correctly

calculating the product of 251 and 864 and solving

quadratic equations with the use of the standard

formula. Towards the end of the book Christopher,

at the age of fifteen, takes an A level mathematics

exam and gains an A grade.

Christopher has a particular interest in prime

numbers which is evident by his choice of chapter

headings which are represented by consecutive

prime numbers 2, 3, 5, 7…

He describes his strategy for identifying prime

numbers (p 14):

This is how you work out what prime numbers are.

First you write down all the positive whole numbers

in the world. Then you take away all the numbers

that is a good thing and good is good enough. For

the moment the target is still there and the evidence

is there to show that both pupils and teachers

are under increasing stress and presenting with

concerning levels of mental health problems in

part due to the ridiculous pressures inherent in

our present education system. I am pleased to

say that many, in fact I would hope most, of my

pupils still enjoy their maths lessons and for me

this is an evaluation of my success as a teacher. I

am struggling to keep stress at bay and enjoy the

experience of being in a classroom and engaged

in learning with my pupils. While my classes are

happy and engaged in their learning then I am

happy that they are making good progress.


that are multiples of 2. Then you take away all the

numbers that are multiples of 3. Then you take away

all the numbers that are multiples of 4 and 5 and 6

and so on. The numbers that are left are the prime

numbers.

Page 14 also contains two diagrams to support the

text. Both of these consist of a 10 x5 array in which

the first consists of all of the numbers from 1 to 49

represented in each of the small squares and the

fiftieth square contains the letters etc. In the second

diagram only the prime numbers are included and

again in the fiftieth square are the letters etc.

Clearly this is an extremely inefficient method

of identifying a prime number. Once all of the

multiples of 2 have been

“taken out” then when

Christopher attempts to

”take out” multiples of 4

he will find that they have

already disappeared!

Hence it is pointless to consider any multiple of

2 to be a prime number. Similarly once all of the

multiples of 3 have been “taken out” it is impossible

for any multiple of 3 to be prime and so on for other

numbers.

Christopher would also encounter considerable

practical difficulties. In the diagrams his numbers

do not go beyond 50. Of course he would need a

much higher number than this if his attempts to

identify a prime number were to be meaningful. It

seems reasonable to say that he would need to go

up to at least four digit numbers. To achieve this he

would need to construct a large number of arrays.

The consequences in terms of the amount of time

and paper that would be required are massive.

This represents a contradiction. On the one hand

Christopher is represented as being a brilliant

mathematician and yet he is seen to use a grossly

inefficient method of identifying prime numbers.

This seriously undermines one of the central tenets

of the book – that a person with Asperger Syndrome

can be a brilliant mathematician and this calls the

whole credibility of the book into question.

It could perhaps be argued that the author has used

Christopher’s convoluted method of identifying

prime numbers as a device to illustrate the fact

that whilst he has outstanding mathematical ability,

it is a consequence of his medical condition that

he is prone to get locked into repetitive methods

of calculation and of

recording data. This

would suggest that he

would have had a poor

examination technique

in which case it is

inconceivable that he would have achieved an A

grade in A level mathematics at the age of 15.

Of course most learners with S.E.N. do not have

Christopher’s outstanding mathematical ability.

A consideration will now be made of resources

and teaching strategies that could be used in an

attempt to enable learners with learning difficulties

to improve their understanding of the concept of

prime numbers.

Strategies aimed at enhancing an understanding

of prime numbers by students with a learning

difficulty.

As a general principle within the teaching of

This seriously undermines one of the central tenets of the book – that a person with Asperger Syndrome can

be a brilliant mathematician


mathematics to students with a learning difficulty

it is helpful to use a variety of different resources

and to use a strategy that is heavily dependent on

repetition and reinforcement through the use of a

variety of different resources and activities. The

following resources should be helpful:

A classroom calendar

1-100 number sheets

Numbered cards from 1-100

Calculators

Classroom calendar

The calendar should be a focal point of the classroom

and consist of colourful, eye catching pictures on a

theme that is of interest

to a particular class such

as pop stars or cats. Each

month would represent

a picture at the top with

individual rectangles for

each of the dates below.

Written entries could then

be made in the appropriate spaces of forthcoming

events such as birthdays of members of the class

or forthcoming school trips. This could then be

used for topical mental maths questions such as

How many days till your birthday? Or How many

weeks till the end of term?

Within the theme of prime numbers, the question

could be Is today’s date prime? Supposing the

date is 28.04.2016 the following strategy could be

used. Ask the class to find the sum of the digits.

When an answer of 23 is given the respondent can

be praised and this answer confirmed as correct.

The class can then be encouraged to check if each

prime number up to 19 is a factor of 19 or not:

2 As 19 is not an even number 2 is not a factor.

3 The sum of all of the digits is 10. 3 is not a factor of 10 and so it cannot be a factor of 19

5 As the final digit is not 0 or 5 then 5 is not a factor.

7 Divide 19 by 7 and a remainder will be obtained so 7 is not a factor.

11 As 2 x 11 =22, 11 is not a factor

13 and 17 clearly are not factors of 19.

19 19 is a factor as 1 x 19 =19.

It can then be explained to the class that it has been

demonstrated that 19 is prime as the only factors

are 1 and 19.

Each member of the

class could then be

asked to calculate if their

birthday is prime or not

and appropriate support

should be provided.

1-100 Number Sheets

Each pupil is provided with a 1-100 sheet and

coloured pencils. The first task is to colour in

blue all of the even numbers, except 2. All of the

multiples of 3(except 3 itself), should then be

coloured in red. These numbers could be found

by starting on 6 and then successively counting

on three squares. Alternatively the constant button

of a calculator could be used by pressing 3 then

+ then 3 and then repeatedly press =. A similar

method could be used to identify multiples of 5

and of 7, except 5 and 7 themselves. Multiples of 5

it is helpful to use a variety of different resources and to use a strategy that is heavily dependent on repetition and reinforcement through the use of a variety of different resources and

activities


can be coloured in green and multiples of 7 can be

yellow. Of course, some squares will contain more

than one colour and this could generate a helpful

class discussion about factors. The discussion

could then move onto a consideration of the reason

why some numbers have not been coloured in. An

understanding that these numbers are prime could

then be encouraged.

Numbered cards 1-100

A game that can be played is called Prime. This is

a game for two players that is similar to Snap. All

of the cards are distributed equally to the players.

Each player then takes

turns to lay a card face

up. If a prime number

is correctly identified the

player who shouts prime first picks up all of the

cards on the table. If the shout is incorrect then the

other player picks up the cards. The game finishes

when one player has no cards left.

Calculators

A variety of teaching strategies can be used with

a jumbo calculator. Reinforcement of multiples

can be used as follows. The calculator should be

positioned such that all of the class or group could

see it. The task could then be to find the multiples

of a particular number. Take the example of 3. The

teacher will slowly press the keys 3 +3. The class

will verbally give their answer and then the equals

button can be pressed to show if they are correct

or not. This procedure can be slowly repeated

with use of the constant button to find successive

multiples of 3.

Once the class are familiar with this activity then the

game Down the Pit can be played. The numbers

1-20 need to be written on a flip chart. Then three

of the numbers at random should be circled. The

class are then asked:

Tell me any number that will not land on any of the

target numbers.

A number will then be suggested. The teacher

then positions the jumbo calculator so that all of

the class can see it. With the use of the constant

button the teacher slowly enters the number and

makes the additions. If a target number is obtained

the pupil who suggested

it goes down the pit. If all

of the target numbers are

avoided then the pupil

does not go down the pit and is praised.

This activity can be linked to the theme of prime

numbers by making all of the target numbers prime.

Useful discussion can then ensue to reinforce the

concept of prime numbers.

Mark Pepper

Note: I was introduced to the game Down The Pit

on a mathematics training course run by I.L.E.A. in

1982ReferencesHaddon,M (2003) The Curious Incident of the Dog in the Night-Time VintagePeacey N (2004) Review in Equals Vol 10 No 2

An understanding that these numbers are prime could then be encouraged.


Maths talk

This edition of Maths Talk features Nichola, a hospital teacher who left mainstream primary teaching 20 years ago.

Could you describe what it means to be a

hospital teacher?

I teach children, from any Key Stage, who have

long-term illnesses. We teach one-to-one, often

at their bedside. The biggest difference is that the

children are ill so I am always aware of their physical

and emotional state at the start of my time with

them. Often their parents and other family members

are present as I teach. It’s a balancing act judging

when a child is ready to learn and when it’s best

to work more on developing a working relationship.

Can you describe some key moments that have

shaped you as a teacher?

In one of my schools I taught tow boys from

Pakistan who had never seen snow before.

We took them outside so they could catch the

snowflakes. That really brought home to me the

importance of experiential learning and the need

to be opportunistic in how you respond to the

needs of your pupils. Now most of my teaching

is opportunistic and I just go with the flow. I feel

comfortable with this because of my experience I

can gently steer the learning down more productive

lines.

Working with children with brain tumors has had a

huge influence on me. It can be challenging as they

regress and so I have to try and help them and their

parents manage the decline without highlighting

it. For example if a child is loosing their sight and

loves working on i-pads then I search for a way

to keep using but also to gradually replace it with

something more closely matched to their abilities.

Mixed ability or setting?

It depends upon the subject perhaps sets can be

more helpful for more academic pupils. I haven’t

taught groups for so long that I do not have a sense

of what work would for me back in the classroom.

You trained at Charlotte Mason College, in

Ambleside. What impact did your four year B.Ed.

training have upon you?

It was the child-centered ethos , espoused by the

lecturers such as Mick Waters and Richard Dunn,

that has stayed with me. The principles I use today

are the same – allowing the children to play a central

role in their own learning. I was also introduced

to High Scope and the plan, do, review process.

Again these same principles underpin my good

practice today. I would love the chance to become

involve din the Forest School movement but that is

not possible with the pupils I now teach.

My first year of teaching was the first year of the

National Curriculum and so I was not trained to

follow the ‘curriculum’ and so it does not come

naturally. If you look at Nursery education and their

pedagogy I feel we would all be so much better off

if that was the pattern all through school, including

A level; we would have purposeful, independent

and happy learners.


How do you approach planning now you mainly

work one-to-one?

I need to get to know the child first by giving them a

range of experiences, games and stories. Only then

do I decide and start my planning, school often

send their plans but we always tailor what we use

to meet the needs of the child. Regardless of my

written plans I always try to follow the lead of the

child. This has enabled me to discover some real

gaps in their knowledge and at times they will share

some of the interesting ideas they have.

What advice would you give to anyone who

wishes to support the lower ability pupils within

their classroom?

Find out what they are interested in and use that.

They are often good at this and feeling positive about

their work really helps in a one-to-one situation. I

do not expect linear progress and feel that naturally

children need lots of repetition, as that is the way

young children learn. This can be hard because

the parents often observe and wonder why they

are doing the same thing over again. I try to make

learning purposeful and fun as I find now that much

of what children do in school can be pretty boring. I

will try to use other, more creative tools, rather than

simply writing. I always allow them to be creative

and to produce something they can be proud of

and then encourage them to share what they have

done with those around them i.e. nurses, Doctors,

parents and their home school.

The teaching and learning of mathematics to pupils with a visual impairment

Mark Pepper has a passion for inclusion, and in the first of a series of articles he outlines the difficulties facing VI students and shares a few strategies their teachers can use to support their learning.

One of the most significant changes in educational

policy in this country in the past twenty years or so

has involved a sharp increase in the number of pupils

with various special educational needs (SEN) being

included in mainstream education. One such group

are pupils with a visual impairment (VI). This change

has resulted in a reduction in the number of special

schools and a dramatic increase in the number of

pupils with a VI being absorbed into mainstream

classes or into VI units within a mainstream school.

Hence a mainstream classroom teacher could have

a pupil with a VI included in their class.

The purpose of this article is to provide suggestions

for appropriate provision to be made to meet the

needs of pupils with a VI. This will include strategies

aimed at minimising the barriers to learning as far

as possible. The initial focus will be on the general

interaction of pupils with a VI with the environment

in their day to day lives and then a more detailed

consideration will be made of their access to the

curriculum generally with a particular emphasis on

the learning of mathematics.


Interaction with the environment

There can be considerable difficulties for pupils with

a VI (especially those who are totally unsighted) in

interpreting the environment. This was forcibly

impressed upon me in the course of a verbal

mathematical assessment I undertook with a Year

11 student who I will call John. Though John has

been totally blind from birth he copes very well in

his everyday life.

One of the questions was:

Paul needs to attach

some numbers to his

front door. He has the

numbers 1, 2 and 3. What

are the possible numbers of his house?

John considered this and then correctly named

all of the single and double digit possibilities. As

I was surprised that he did not suggest any 3 digit

numbers I asked him if he could think of any other

answers. He responded by saying that he knew

there couldn’t be any 3 digit numbers. When I

asked for his reasons he replied that “no road could

be that long.”

Upon further discussion it became evident that

John thought that all

roads were short. As he

had never seen a road

this was a reasonable

assumption. He showed

considerable surprise

when I explained to him that most roads consisted

of hundreds of houses.

Hence John had not made a mathematical error

despite the fact that he didn’t provide all of the

possible door numbers. If less time had been

available and John’s answer had been accepted

without discussion he would have been incorrectly

assessed as having a weakness with the use of 3

digit numbers.

The use of touch

For the pupil with a VI there is a heavy dependence

on the use of touch to compensate for the loss of

sight. Hence skills have to

be developed to facilitate

the ability of the pupil

to be able to interpret a

brailled text by carefully

feeling the raised lettering of the braille. Similarly

tactile diagrams would need to be accessed

through the medium of touch.

Advances in Technology

Recent advances in technology have resulted in

greatly improved learning resources for learners

with a VI. Perhaps the first of these advances

was created with the introduction of software,

such as Jaws and Super Nova, that consisted of

magnification and screen readers. This meant

that an automated voice,

invariably a male voice

with a strong American

accent, would read aloud

the text that appeared

on the computer screen.

In recent years associated conferencing software

technology such as Display Note and Team Viewer

have been developed to such a sophisticated degree

John thought that all roads were short. As he had never seen a road this was

a reasonable assumption.

He showed considerable surprise when I explained to him that most roads consisted of hundreds of

houses.


that an electronic link can be made between the

learner’s iPad and both the teacher’s computer and

the interactive whiteboard. This innovation enables

a learner to independently access and enlarge the

image or text on the whiteboard or the teacher’s

computer. The iPad can also be used to magnify

texts and diagrams from

books or worksheets

to an appropriate size.

An impressive resource

for braille users consists of the braillenote. This

is a device that is smaller than an iPad. It has a

brailled keyboard and the brailled output can be

converted into speech. A connection with the

internet can also be made and other devices such

as a talking calculator can also be installed on it.

Easy converter software is also available and this

device can convert print into speech, large print or

braille. Another recent advance has involved the

introduction of portable CCTVs. Formerly a CCTV

would be statically located in the classroom for use

by learners with a VI. The portable CCTV enables

the learners to take it

with them as they move

around the school. Some

of them even have an

inbuilt high definition

distance camera in order to view the whiteboard,

learning walls and the teacher demonstrations.

Making the curriculum as accessible as possible

A key requirement for a class teacher who has a

learner with a VI in her/his class is to make the

curriculum as accessible as possible to that child.

Extremely valuable support can be obtained by

contacting the L.E.A. Visual Impairment Service.

Support can then be provided on a wide range

of issues that could include adaptation to the

classroom environment, mobility, one to one

support, resources and the adaptation of resources.

Adaptations within the classroom

There are two main

categories of learners

who have a VI. One of

these consists of learners

who have sufficient sight that they can access

enlarged print whilst the other category consists of

learners with very limited sight who use braille. The

former will be referred to as print users whilst the

latter will be referred to as braille users.

Print users

If the lighting is considered to be inadequate then a

light can be provided at the learner’s desk .

In some cases a sloping desk can be appropriate.

The use of CCTV can be beneficial though a

portable CCTV would be more efficient.

It can be helpful to make

appropriate magnifiers

available.

The learner should

be positioned at an

appropriate distance from the whiteboard to be

able to access the information on it.

Braiile users

Additional lighting can be helpful for a learner with

some sight.

In some cases a sloping desk can be appropriate.

Mobility

Braille users

An impressive resource for braille users consists of the braillenote.

Another recent advance has involved the introduction of portable CCTVs.


It is essential that the learner becomes confident

in accessing different parts of the school. The VI

service can provide training such that the learner

memorises the routes to other parts of the school

and within the classroom is able to independently

locate resources.

One to one support

Print users

Individual support can be provided by a VI tutor but

is more commonly provided by a teaching assistant

attached to the school and this could well only be

available on a part time basis. The support can

involve verbal advice on a text or diagram that the

learner has difficulty in seeing in detail. Written work

can be recorded for the learner with the use of a

thick black pen with a contrasting background such

as white. In some instances an individual learner

may show a preference for a different background

colour.

Braille users

The individual support for a braille user can vary

considerably in accordance with the different needs

of learners. If the learner is not an accomplished

braillist then the support

could focus primarily

on the brailling of

information. It could

also be the case that

considerable support

would be required in the interpretation of tactile

diagrams.

In instances where the learner is an accomplished

braillist then verbal descriptions of objects or

diagrams would be appropriate.

Adaptation of materials

Print users

Resources such as worksheets should be produced

in enlarged form to a size that is in accordance

with the learner’s visual

assessment . A visual

assessment can be

undertaken by the VI

Service.

Diagrams should consist of thick black lines on an

appropriately contrasting background.

A laptop computer should be made available.

A large key calculator with a large display should be

made available.

Braille users

Written materials should be in a brailled format and

diagrams should be in tactile form.

Equipment such as rulers should be in tactile form.

IT Resources

Print users

An iPad as described in Advances in Technology

section should be provided.

A calculator with large

buttons and a large

display should be made

available.

Braille users

A laptop computer together with a software

programme such as Jaws or Super Nova should

be provided.

Audio tapes would be a helpful resource.

A talking calculator should be made available.

The individual support for a braille user can vary considerably in accordance with the different needs of learners.

Written materials should be in a brailled format and diagrams should

be in tactile form.


Funding

The effects of current Government policies and the

present economic climate place great restraints on

school budgets. Whilst the above recommendations

represent the resources

that would ideally be

required some schools

may be unable or

unwilling to finance these resources, especially the

IT hardware. In such cases it is to be hoped that at

least some of these needs would be met.

With reference to payment for services from the LEA

VI Service the usual practice is for no charge to be

made to schools as they would already have made

a contribution to the LEA for generic needs. There

are, however, variations between different LEAs

regarding academies. Some LEAs currently do

not charge academies if the child who is receiving

support resides in the LEA. Other LEAs do charge

academies on the basis that they would not have

made any contribution to LEA funding.

Integration

One of the main purposes of educating learners with

a VI within the mainstream system is to encourage a

policy of inclusion. Hence learners from a variety of

backgrounds and with different needs can socialise

amicably. The fact that this can work efficiently was

demonstrated when I visited a mainstream primary

class that included a girl with a VI, who I will call

Natalie. A maths problem

had been written on a flip

chart and the teacher

asked for a volunteer to

come and write the answer. Natalie volunteered

and as she approached the flip chart one of her

classmates said:

“She can do it. Its only her eyes that don’t work

well.”

That comment effectively encapsulated the

co-operative spirit that existed in the class.

Mark Pepper

Mark Pepper was Head of Mathematics at Linden

Lodge School for blind and visually impaired pupils

from 1996-2006

I would like to extend my thanks to Tim Richmond,

VI Specialist Teacher at Wandsworth Vision Support

Service, who has provided helpful advice on current

issues within the education of learners with a VI.

“She can do it. Its only her eyes that don’t work well.”


Mikos De Symmetria: a special approach to Trigonometry

We were approached by a young mathematician from India, Sameer Sharma, who sent a really interesting article on Trigonometry. We felt that we could do little else but showcase the thinking he has been doing and if you would like to share any thoughts with him please e-mail the Editor who will pass them on.

I am 14 years old and in grade 1X. My favorite

subjects are Mathematics and Science.

In my opinion mathematics is something from

which all the major subjects are related for example,

physics related with arithmetic, construction work

(buildings etc.) is related with geometry and many

more……

In future, my aim is to become a Research Scientist

at ISRO. And I will make Research in Mathematics

as my passion.

Name: Mikos De Symmetria

The name Mikos de Symmetria has been derived

from the Greek words “Mikos” Meaning length and

“Symmetria” meaning measurement. Which means

Length Measurement through Scale.

Introduction

As we all know that Mathematics plays a very

important role in our life. By Trigonometry we can

find the length of any building, object etc. using the

ratios of angles and their sides.

Being very impressive in nature, it also has many

drawbacks some of them are as follows:

1. Its Instrumentation is very complicated and also

very expensive.

2. Trigonometry works only when the object is

perpendicular to the surface.

3. It has around 2160 Ratios which are difficult to

learn and remember.

In order to overcome the above mentioned

drawbacks, I have carried out another method named

“Mikos de Symmetria” also its instrumentation is

very simple to make and use.

Scale:

Being very simple in nature. It is the one of the most

powerful mathematical tool since it has the ability

to convert in any other mathematical instrument

such as protractor, compasses etc. the only thing

we need is our IMAGINATION. In my opinion it is

the one of the most powerful tool ever made which

I have seen until now.

Instrumentation

Materials required:

1. A flat board

2. A clean Scale (in centimeters)

3. A base Stand of certain height.

4. Moveable rotator

5. Magnifying glass (optional, just for more

accurate measurements.)


In order to make the apparatus, take the base and

stick the board on it. And on the board, stick the

scale headed by moveable rotator.

Main concept

The main concept behind this research is:

“THE FURTHER AWAY THE OBJECT IS, THE

SMALLER IT SEEMS TO BE AND VICE VERSA”

Experiments

I have done various experiments in order to find the

relationship between scalar height (i.e. The height

which is measured by scale) and the actual height.

They are done in the following way.

In doing the experiments we have to find the scalar

lengths from a place. Let’s see some illustrations:

(see Figure 2 below)

OBSERVATION – I

On the basis of illustrations, I have made the table

(Table 1) measuring various objects. These include

the following objects:

1. Bottle cap

2. Bottle

3. Watch box

4. Flash drive

5. Cardboard box

Measurements Taken:

1. At 30 Cm away.

scale

Magnifying Glass Rotator

Base

Stand

Figure: Instrumentation

Figure 2

Eye

Scale

Scaler Length

Distance Distance

Box

X units Y units

Table 1

Objects Actual Length At 30cm At 15cm Length

Observation - I Observation - II Increased

Bottle cap 2.6 cm 1.4 cm 1.8 cm 0.4 cm

Bottle 9.0 cm 4.8 cm 6.2 cm 1.4 cm

Watch Box 6.3 cm 3.0 cm 4.1 cm 1.1 cm

Flash Drive 2.6 cm 1.4 cm 1.8 cm 0.4 cm

Box 3.0 cm 1.5 cm 2.0 cm 0.5 cm


2. At. 15 Cm away.

Distance between eye and scale is 29 cm.

Derivation of the formula

Let the object of height ‘h’ is placed at a distance of

k units away from scale and distance between the

scale and the eye be y units.

Let us represent this illustratively: (see Figure 3

above)

From the above illustration we see that two triangles

are formed (See Figure) i.e. ∆ ABC and ∆ ADE

Now in ∆ ABC and ∆ ADE

∠ BCA = ∠ DEA

∠ BAC = ∠ DAE

Therefore, by AA similarity Criterion,

∆ ABC is similar to ∆ ADE, since they will form same

ratio,

Therefore, AE = DE

And in the above equation, we know AE, EC, DE.

By putting the values of three variables, we will get

the formula as:

Actual length (BC) = (EC × DE)

Hence Derived!

Importance of this research

It has many significant factors:

1. It is easier than trigonometry.

2. There is no effect of angles on it.

3. It can even find the length of one which is not

perpendicular to its surface.

4. It can find the diameter, surface area, volume

of moon and even stars!!!!

Precautions while carrying out experiments:

1. Head should be kept in same position means

there should be no shift in eyes while observing

the object.

2. Scale should be neat and clean means the

markings should be clearly visible on it.

Figure 3

Figure 3Eye

Scale

Scaler LengthD

A

E C

B

Distance Distance

Box

X units Y units

EC BC

AE

Ruler with Rotator

Magnifying Glass

Box


Illustrations

Suppose we want to find the length of building,

then the correct way of measurement will be: (see

Figure 4 below)

The illustration below is the correct way of

measuring objects.

Special cases:

1. Slant case:

For the object which are not perpendicular to the

ground, it will be difficult to find their lengths by

trigonometry, but their lengths can be easily be

found by MIKOS DE SYMMETRIA. For more details,

let us see the illustration….

2. Celestial case: case of moon

We can also find the measure of the celestial

bodies using this formula which cannot be found

using Trigonometry. Let us understand it through

Illustrations. As we all know that moon revolves

round the earth on an elliptical orbit. That means

there are two points which are the farthest point

and the nearest point. The two points are known as:

Closest Point (PERIGEE) = 225,623 Miles

Farthest Points(APOGEE) = 252,088 Miles

Magnifying Glass

Figure 4

Figure 5: Measuring objects with slanted heights

Object

Distance: object-scaleDistance: eye-scale

Scale

X’ unit

Y’ unitK unit

Eye

Base

Slant

Scale

Eye

Stand

Surface

Rotator


These two points can be identified using the phrases

of the moon as the quarter part is the PERIGEE and

the full moon or the new moon is the APOGEE. Let

us understand it through Illustrations…….

Conditions for this method:

1. We should know the distance between object

and scale no matter how we measure the distance.

we can even measure the distance with our foot.

Epilogue

At last I want to conclude that by using this method

we can find the length of any object using the

relationship between scale-object distance and

scale-eye distance making it easier to use than

trigonometry and the formula used is very simple

to remember than trigonometric ratios. So I will

consider it as easier method of trigonometry.Figure 6: Image showing apogee and perigee

Objects to think with. Papert (1980)

In this piece Mary Clarke reflects upon the importance of the work of Papert and how his ideas might help classroom teachers today.

Making the Most of Manipulatives

One of the most effective ways to help children

really understand maths is to use a wide variety

of manipulatives. Manipulative materials are any

objects that pupils can touch move or arrange to

model a mathematical idea or concept. They help

children to explore the patterns and relationships

in maths in a concrete way. However, their use

is nothing new. Piaget in 1951 recommended

manipulatives as a way of embodying mathematical

concepts. Gattegno and Cuisenaire developed

Cuisenaire rods in 1954 and Dienes developed

his base ten materials in 1969 as a way of helping

children to grasp fundamental mathematical

concepts.

In the 1960s Bruner put forward the idea that

children need to go through three stages, concrete-

pictorial-abstract, in order to fully understand a

mathematical concept.

Briefly his model can be described like this:

Concrete. At the concrete level manipulatives are

used to explore and solve problems.

Pictorial. At the pictorial level pictures, drawings,

diagrams, charts, and graphs are used as visual

representations of the concrete manipulatives.

Abstract. At the abstract level symbolic

representations are used to represent and solve the

problem.

This is the basis for some of the most successful

methods of teaching maths, including the Singapore


approach and the approach used by researchers

such as Professor Sharma in supporting children

with dyscalculia.

Stein and Bovalino (2001) stated

“Manipulatives can be important tools in helping

students to think and reason in more meaningful

ways. By giving students concrete ways to compare

and operate on quantities, such manipulatives as

pattern blocks, tiles, and cubes can contribute to

the development of well-grounded, interconnected

understandings of mathematical ideas.”

So it is clear that manipulatives are an essential

tool for teaching mathematical concepts. They

help children to understand the maths and develop

visualisation skills which are so important for making

sense of maths. However,

many practitioners are

unclear of how to get

the most out of the

manipulatives they have

in their classrooms and

many children simply use them as a crutch or tool

to support a particular mathematical procedure.

Ofsted’s 2012 report ‘Made to Measure’ suggests

that although manipulatives are used in some

primary schools to support teaching and learning

they are not used as effectively or as widely as they

might be.

This means that the children are not getting

the depth of understanding that they could be

achieving through a more consistent and effective

use of manipulatives. Using a wide variety of

manipulatives can help children to develop a range

of skills and concepts including:

• Sorting

• ordering

• spotting patterns

• recognising geometric shapes

• understanding base-ten and place value

• the four mathematical operations— addition,

subtraction, multiplication, division

• exploring and describing spatial relationships

• developing and using spatial memory

• problem-solving

• representing mathematical ideas in a variety of

ways

• making connections between different

concepts

• communicating mathematical ideas effectively

A key point is to make sure that the children have a

wide variety of manipulatives available, for as long

as they need them. Using

manipulatives over long

periods of time is much

more beneficial than

occasional short term

use.

Children who habitually use manipulatives make

gains in the following skills

• using mathematical language to explain their

thinking

• relating real-world situations to abstract maths

• working collaboratively

• thinking more flexibly to find different ways to

solve problems

• developing metacognition

• working independently

• being more resilient and persevering with

problems

One particular problems that children have when

pattern blocks, tiles, and cubes can contribute to the development of well-grounded, interconnected understandings of mathematical

ideas

learning maths is that they see each area of maths

as separate and unconnected. For example,

division can be taught as repeated subtraction

but children may also come across fractions in

terms of shading segments of a cake and they are

unable to make the link between the two concepts.

This compartmentalising of maths can be very

detrimental to children who are struggling and have

poor number sense. One way to overcome this is to

highlight patterns and connections between areas of

maths by using continuous

manipulatives such as

base-10 materials/Dienes

and Cuisenaire rods.

Initially children will often prefer to work with discrete

materials such as counters, beads or cubes, but it

is a good idea to move on from these materials so

that they can move on from counting in ones. Using

a variety of manipulatives will help the children to

make links between areas of maths and will help

them to see it as an interconnected rather than

compartmentalised subject.

The use of manipulatives has been shown to support

children who are struggling as well as extending

more able children. It also

makes for a more inclusive

classroom environment

and helps to reduce maths

anxiety. Above all, maths

is much more fun for the

children if they have a variety of manipulatives at

their fingertips.

So how can we best use manipulatives in the

classroom?

Common pitfalls

Children using the manipulatives as crutches to

follow a rote procedure rather than as a means to

developing understanding.

Teachers selecting a certain type of manipulative to

teach a particular concept.

Using manipulatives as an add on to a lesson or as

a reward.

Using manipulatives only for children who are

struggling.

One of the greatest causes

of maths anxiety in the

classroom comes from

manipulatives being taken

away too soon, leading to

maths becoming abstract and meaningless.

The way forward

Free play, letting the children discover the properties

of the manipulative for themselves.

Access to a wide variety of manipulatives, with

children being able to choose what they want to

work with.

Manipulatives readily available from Foundation

stage to Key Stage 3 and preferably beyond.

Encouraging children to

demonstrate the idea with

their chosen manipulative.

Encouraging children to

generalise.

Ask the children to show you and each other

different ways of solving a problem using a variety

of materials.

(This article is an extract taken from ‘Making the

Most of Manipulatives’ by Judy Hornigold. Due to

be published by SEN Books in March 2016)

One particular problems that children have when learning maths is that they see each area of maths as

separate and unconnected

Initially children will often prefer to work with discrete materials such as

counters, beads or cubes


Review - Rachel Gibbons

Embedding Formative Assessment

Dylan Wiliam & Siobhan Leahy

Learning Sciences International

Although this book is written very much in the

context of the American classroom, it contains

a wealth of novel ideas for making effective

formative assessment which will fit just as well to

the classrooms of this country as American ones.

Wiliam and Leahy have

many years of experience

teaching in schools on

both sides of the Atlantic

- and indeed also in

leading and training teachers in the UK and the

USA.

It is always difficult to make assessment really

meaningful to a class: the pupils who have been

successful in the work they have done are eager

to press on to fresh fields rather than to look back

to see what could have been improved in work

already completed, whereas those whose work

has proved inadequate

do not necessarily want

to be reminded of their

failure. I have observed

too many lessons “going

over homework” which has been a complete waste

of time for the majority of the pupils in the class.

But so much can be learnt from a study of work

already completed that it is important to make

some attempt at formative assessment. Formative

assessment is indeed vital to effective teaching.

For how else, as Wiliam and Leahy point out, can

we know where to l lead our pupils next in their

mathematical journeys? One solution Wiliam and

Leahy suggest to tackle the difficulty of getting

pupils to look back which particularly appealed to

me is to write your assessments of individual

pieces of work, not in pupils’ exercise books, but on

separate slips of paper.

Then hand back a set of,

say, five books with the

five relevant assessments

written on separate slips

and ask the group of

pupils to fit the assessments to the relevant pieces

of work. What a valuable discussion must arise

from that task: discussion based on a thorough

examination of the five versions of the completed

task. I wish I had thought of that – so simple yet so

effective!

For any teacher examining the relationship

between what she is attempting to teach and

what her pupils have

learnt is vital as Wiliam

and Leahy remind us for

students do not always

learn what we teach, and

we had better find out what they did learn before

we try to teach them anything else - which is why,

of course, these authors stress the importance of

formative assessment being firmly embedded in all

our teaching.

It is always difficult to make assessment really meaningful to a

class

these authors stress the importance of formative assessment being firmly

embedded in all our teaching

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